While I'm not experienced, my guess is in 5 minutes you can't delta hedge enough times to be able to be anywhere close to risk free i.e. you can't expect realized vols to equal the implied vols in such a short period.
So in my mind as long as no arbitrage conditions are satisfied I would price based on supply/demand/market speculation rather than replication. One solution is to raise the implied vol quite high to compensate for the now unhedgeable PnL variance. You can raise it such that say the 10th pecentile of your PnL distribution is greater than some tolerance level.
Edit: Bit of formalism to address the comment:
The delta hedge done once leaks a PnL:
But if we delta hedge multiple times over the life of the option, the net PnL is:
which is sum sort of a weighted average of the difference between implied vol (fixed) and realized vols (random variables). If chosen correctly, sum of realized vols converges to the implied vol in a central limit theorem style (ignoring for once the complexity of gamma weights, and vol autocorrelation).
In this case PnL variance due to discrete hedging is essentially the "standard error" of the convergence procedure. So the more times you delta hedge the better convergence you can get.
In this instance you possibly delta hedge only once so the PnL variance is massive. "n" of the central limit theorem is small. So you can't really expect any meaningful hedging. Also therefore idiosyncratic moves in the stock (which otherwise average out over multiple delta hedges) now completely control your PnL in line with dm63's first comment.
Not to mention that gamma of the option is also quite large close to expiry. Therefore my comment that the product has to be priced with how much "comfort" you have over the PnL variability.